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**Statistical Mechanics and Evolutionary Theory**

Lloyd Demetrius Harvard University, Cambridge, Mass., USA And Max Planck Institute, Berlin, Germany

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**Evolutionary changes in morphological complexity**

Ecological time scale (Single evolving lineage) Increases and decreases in adult body size Geological time scale (Phyletic lineages) Increases in mean body size Geological time scale (Clades) Increases in maximum body size

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**Evolution of the horse family**

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**Changes in body size within the equid lineages**

Increase in body size : North America Decrease in body size : Europe

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**Increase in mean body size within the equid taxon**

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**Increase in maximum body size over the history of life**

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Problem What is the evolutionary basis for the changes in body size over evolutionary time ?

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Darwinian argument Individuals differ in terms of their morphology, behavior and other phenotypic characteristics (variation) Different phenotypes are characterized by differences in the acquisition and transformation of resources (natural selection) There exists a correlation between the characteristics of parents and their offspring (heredity)

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Darwinian fitness The efficiency with which organisms transform resources into net offspring production

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**Levels of biological organization**

Populational: Changes in the phenotypic composition of a population by a natural selection Phyletic lineage: Changes in the species composition of a lineage by speciation and background extinction Clade: Changes in the species composition of a clade by speciation and mass extinction

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**If W2 > W1 : then X2 replaces X1**

Darwinian model Organic diversity and changes in complexity can be explained in terms of the following tenet Selection tenet Resident type X1 ; Fitness W1 Variant type X2 ; Fitness W2 If W2 > W1 : then X2 replaces X1 Fitness The efficiency to transform respurces into net-offspring production X1 X2

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Darwin‘s theory Evolutionary Principle: Evolution by natural selection results in an increase in fitness Explanatory Power Variation in life history, body size, life span within and between species The adaptation of species to their habitat The changes in morphological complexity over time

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Problem Can Darwin‘s argument be translated into an analytical theory which will explain: The diversity of species in space and time The adaptation of species to their environment The increase in complexity within lineages

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Does there exist a demographic characterization of fitness which will predict the outcome of competition between variants and incumbents in a population of organisms ?

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**Characterizations of Darwinian Fitness**

Malthusian parameter (1930) Fisher‘s theory Evolutionary entropy (1974) Directionality theory

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**The theory of evolution by natural selection**

is the doctrine of Malthus applied to plants and animals. Darwin (1859)

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**Population described by d age-classes**

Demographic model Population described by d age-classes bi = Probability of surviving from age-class (i) to age-class (i+1) mi = Mean number of offspring produced by individual in age-class (i) lj = b1,b2,...,bj-1 = Survivorship to age-class (j) Vj = lj mj = Net-reproduction at age j

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**Malthusian parameter as Darwinian fitness**

Matrix Representation of Graph Characterization of r :

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**Growth rate r characterizes Darwinian Fitness:**

Fisher‘s Theory Growth rate r characterizes Darwinian Fitness: Malthusian Principle: r predicts the outcome of competition between variant and incumbent types r r* r X* X X* X r*

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**Fisher‘s evolutionary theory**

Population growth rate Mean Fitness Fisher‘s principle: Evolution by natural selection results in an increase in the mean malthusian parameter

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**The Malthusian Parameter as Darwinian Fitness Critique**

Computational studies: In Competition between mutants and the resident population the growth rate is not always a good predictor of invasion success Empirical studies: Invasion success is highly correlated with body size and is contingent on the resource constraints

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Darwin‘s theory of evolution by natural selection is the doctrine of Gibbs, Boltzmann and Clasius applied to plants and animals. Directionality theory (1974)

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**Directionality theory**

Evolutionary entropy, S , characterizes Darwinian Fitness

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**Evolutionary principles**

Evolutionary dynamics within a single evolving lineage (Mutation and Selection) Directionality Principle for Entropy Limited Resources: Evolution increases entropy Variable Resources: Evolution decreases entropy Evolutionary dynamics within a taxon (Speciation and Extinction) Fundamental Theorem of Evolution The rate of change of mean entropy is equal to the variance in entropy Mean entropy increases over geological time Evolutionary dynamics within a a clade ( Speciation, background and mass extinction ) Secondary Theorem of Evolution The upper entropic limit of species in a clade increases as the claded replaces another over geological time

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**Organization The origin of evolutionary entropy: Its demographic basis**

The directionality principles for evolution: Their mathematical basis Implications of directionality theory for the study of Life history evolution Evolution of body size Evolution of senescence

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**Origin of evolutionary entropy Demographic model**

Microstates: Population growth rate:

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**Biological networks Macrostates from microstates**

P Ann. App. Prob. (1974) P 3.

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**Demographic networks Macrostates from microstates**

Entropy: Reproductive potential: Generation time:

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**Properties of entropy Measure of uncertainty Measure of diversity**

Measure of robustness

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Uncertainty measure Uncertainty in the age of the mother a randomly chosen newborn pj Probability that the mother of a randomly chosen newborn belongs to age class (j)

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**Robustness Genealogies: Set of paths of the graph Path:**

Matrix associated with the graph 1 3 2 d ...

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**Robustness Theorem: Annals. App. Prob.(1994)**

Prob. that the sample mean Theorem: Annals. App. Prob.(1994)

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**Reproduction potential and resource constraints**

Proposition: In Populations in dynamical equilibrium with resource conditions E<0: Constant resource E>0: Variable resource

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**The Entropic Selection Principle**

Entropy as darwinian fitness Competition betweem variant and incumbent is a stochastic process determined by entropy (S) and contingent on the resource constraints (E) Limited resources: (E<0) Mutants with increased entropy have increased robustness and will prevail (a.s) Variable resources: (E>0) Large population size: Mutants with decreased entropy will have decreased robustness and will prevail (a.s) Small population size: The outcome of competition will be a stochastic process described by probabilities contigent on population size

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**Invasion dynamics Evolutionary entropy predicts the outcome of competition**

Limited Resources X X* S S* X X* S* S Variable Resources X X* S S*

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**Predictions of directionality theory**

Based on the entropic principes of selection we predict the evolutionary changes at three different levels of biological organization. Single evolving lineage – Mutation and selection Aggregate of phyletic lineages – Speciation and background extinction An ensemble of clades – Speciation and mass extinction

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**Evolutionary dynamics within an evolving lineage**

Long run changes in entropy as one population type replaces another under mutation and natural selection Equilibrium species: Species subject to limited resource conditions Opportunistic species: Species subject to variable resource conditions Evolutionary principles: Entropy increases in equilibrium species Entropy decreases in opportunistic species

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**Evolutionary dynamics within a taxon**

Long run changes in mean entropy as one phyletic lineage replaces another under speciation and background extinction. The rate of change in mean entropy is equal to variance in entropy Mean entropy increases

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**Evolutionary dynamics within a clade**

Long run changes in maximum entropy as one clade replaces another under mass extinction The upper entropic limit increases as one clade replaces another over geological time.

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**Main tenets of the evolutionary process**

Evolutionary dynamics within a single evolving lineage Equilibrium species: Entropy increases Opportunistic species: Entropy decreases Evolutionary dynamics within a taxon The rate of change of mean entropy is equal to the variance in entropy Evolutionary dynamics within a clade The upper entropic limit increases as one clade replaces another

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**Implications of the evolutionary tenets**

Evolution of life history Evolution of body size Evolution of senescence

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**Body size and physiological time**

Allometric relations Body size and physiological time Physica A. (2003) Physiological time, Body size Physiological time Cycle time of metabolic processes Generation time Life span

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**Entropy and generation time**

Theorem

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**The evolution and distribution of species body size**

Relation between entropy S and body size W

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**Empirical study Relation between entropy and body size**

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**Directionality theory predicts evolutionary changes in body size**

Changes in body size within a single evolving lineage Limited resource conditions Increase in body size Variable resource conditions Decrease in body size

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**Changes in body size within the equid lineages**

Increase in body size : North America Decrease in body size : Europe

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**Directionality theory predicts evolutionary change in body size within a taxon**

The rate of change of the mean body size of species within a phyletic lineage is equal to the species variance in body size Mean body size increases within a taxon ( Cope‘s Rule )

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**Increase in mean body size within the equid taxon**

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**Evolutionary changes in the upper limit of bodysize**

The upper limit of body size increases as one clade replaces another over geological time.

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**Changes in the upper limit of body size**

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**The evolution of life span**

Evolutionary entropy is analytically related to life span L Directionality theory predicts species variation in life span

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**Empirical observation Entropy and life span**

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**The evolution of senescence**

Directionality theory explains variation in the rate of aging between equilibrium and opportunistic species. Proposition: The intensity of natural selection is a convex function of age

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**Intensity of natural selection**

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**Conclusion Darwinian Fitness is characterized by evolutionary entropy**

Diversity of species and evolutionary change in complexity can be described in terms of the following tenets: Population level: Equilibrium species: Entropy increases Opportunistic species: Entropy decreases Phyletic level: Mean entropy increases Clade: The upper entropic limit increases

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**Relation between thermodynamic variables and evolutionary parameters**

Free energy, Thermodynamic entropy, Temperature, Mean energy, Evolutionary parameters Growth rate, Demographic rate, Reciprocal generation time, Reproductive potential

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**Relation between thermodynamic principles and evolutionary principles**

Thermodynamic entropy: Diversity of energy distribution Demographic entropy: Diversity of energy flow The entropic principle for evolution is a non—equilibrium analogue of the entropic principle for physical systems.

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**Relation between thermodynamic principles and evolutionary principles**

Thermodynamic entropy: Demographic entropy: Analytic relation between generation time, and Temeprature : Theorem: The entropc principle for thermodynamic systems is the limit of the entropic principle for evolutionary processes.

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